In this paper we consider billiards in a square with a barrier. We use Ratner's theorem to compute the asymptotics for the number of closed orbits.
Alex Eskin. Howard Masur. Martin Schmoll. "Billiards in rectangles with barriers." Duke Math. J. 118 (3) 427 - 463, 15 June 2003. https://doi.org/10.1215/S0012-7094-03-11832-3